ACJC 2025 JC2 H2 Prelim Paper 2 QP

ANGLO-CHINESE JUNIOR COLLEGE JC2 PRELIMINARY EXAMINATION Higher 2 MATHEMATICS 9758/02 Paper 2 29 August 2025 QUESTION PAPER 3 hr Additional Materials: Printed Answer Booklet List of Formulae and Results (MF27) READ THESE INSTRUCTIONS FIRST Answer all questions. Write your answers on the Printed Answer Booklet . Follow the instructions on the front cover of the answer booklet. Give non-exact numerical answers correct to 3 significant figures, or 1 decimal place in the case of angles in degrees, unless a different level of accuracy is specified in the question. You are expected to use an approved graphing calculator. Unsupported answers from a graphing calculator are allowed unless a question specifically states otherwise. Where unsupported answers from a graphing calculator are not allowed in a question, you are required to present the mathematical steps using mathematical notations and not calculator commands. You must show all necessary working clearly. The number of marks is given in brackets [ ] at the end of each question or part question. _________________________________________________________________________________ This document consists of 8 printed pages. [Turn over

2 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 Section A: Pure Mathematics [40 marks] 1 It is given that 3 + 3OA += i j k and 5 4 3OB −+= i j k . Find the position vector of a point R on line OB such that AR is perpendicular to OB. Hence find the position vector of the point A, the reflection of the point A in the line OB. [4] 2 It is given that 2 1f( ) 14 x x = − where 11 22 x−   . (a) Using standard series from the List of Formula e (MF27), find the Maclaurin’s expansion of f( )x , up to and including the term in 6x . [2] (b) Hence find the first four non-zero terms of the Maclaurin series for 1sin 2 x− . [4] 3 (a) For this question, you may use these results: ( )( )2 1 1 2 1 6 n r n n nr = ++= and ( )3 22 1 4 1 n r nnr = += . (i) Show that ( ) ( )( )( )2 1 1 2 3 51 12 n r n n n nrr = + + ++= . [3] (ii) Hence find ( )( ) 1 2 5 2 3 n r r r − = + + in terms of n. [3] (b) The sequence 1 2 3, , ,u u u is defined by 1 2u = , 1 1 ,11 n n un u + =− . Find the values of 2u , 3u and 4u . Hence find the value of 2025u . [3]

3 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 [Turn over 4 The function f is defined by 2 1f: 65x xx++ , for 3x− , 1x− . (a) Find ( ) 1f x− . [3] (b) Find algebraically the range of f. [4] The function g is defined by g : e xx , for x . (c) Find the exact range of gf. [2] 5 (a) By using the substitution tanxa = , show that 2 2 2 2 11 d ln xxc ax x a x a a =+ + + +   , where 0x and 0a . [4] (b) At time t in a chemical reaction, x kg of substance X and y kg of substance Y are present. Initially, there is 4 kg of X and 5 kg of Y . The variables x and y satisfy the equations d d x xyt = and 2d d y xt = . (i) Find d d y x in terms of x and y, and by solving this differential equation, show that 2 9yx=+ . [4] (ii) Obtain a differential equation in terms of x and t only and hence find an expression of t in terms of x. [4]

4 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 Section B: Probability and Statistics [60 marks] 6 (a) If P( | ) 0.2BA = , P( | ) 0.6AB = and P( ' ') 0.3AB= , find P( ).AB [3] (b) The events X, Y and Z are such that the events X and Z are mutually exclusive. Given further that P( ) 0.1XY= , P( ' ) 0.35XY= , P( ) 0.2YZ= , and P( ) 0.95X Y Z  = , find the minimum and maximum values of P( )X . [3] 7 A gathering of 12 people break into 3 groups of 4 to play 3 different games: carrom, UNO and bridge. (a) Find the number of ways these 12 people can be grouped. [1] (b) Aaron and Sandy do not get along with each other and do not want to be in the same group. Find the number of ways these 12 people can be grouped with this restriction . [3] (c) Given that Aaron and Sandy are not in the same group, find the probability that a third person Billy is in the same group as Aaron. [3] 8 The probability distribution of a discrete random variable X is given as follows: Given that E(X) = 0.9 and Var(X) = 2.99, find the values of a, b and c. [5] x –2 –1 0 1 3 P( )Xx= 0.1 a 0.3 b c

5 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 [Turn over 9 A fruit seller sources his oranges from a supplier with a large plantation where 1% of the oranges are rotten. The supplier picks the oranges regardless of whether the oranges are rotten or not, and packs 20 oranges in each box at random. (a) The random variable X denotes the number of rotten oranges in a randomly chosen box. State in context two assumptions for X to follow a binomial distribution. [2] Assume that X follows a binomial distribution for the rest of this question. The fruit seller considers a box containing between 2 and 5 rotten oranges inclusive as substandard. (b) Find the probability that a randomly chosen box is substandard. [1] (c) Find the expected number of rotten oranges in a box. [1] A truckload of 10 randomly chosen boxes of oranges is then sent to the fruit seller. (d) Find the probability that not more than 2 boxes from the entire truckload are substandard. [2] Upon receiving the oranges, the fruit seller inspects the boxes with these conditions: • Accept the entire truckload if the first box opened has no rotten oranges. • Reject the entire truckload if the first box opened has more than 1 rotten orange. • If the first box has exactly 1 rotten orange, the fruit seller opens a second box. • If the second box has at most 1 rotten orange, he will accept the entire truckload. Otherwise, he will reject the entire truckload. (e) Find the probability that he will reject the entire truckload. [2]

6 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 10 Scientists are studying the growth of a particular species of bamboo. They want to understand how the height of the bamboo stalk changes over time. The table below shows the height, h inches, of a bamboo stalk at age, t weeks, after sprouting. (a) Draw a scatter diagram of these data, and explain using the diagram why the relationship between h and t is not well -modelled by an equation of the form h a bt=+ or 2h a bt=+ , where 0b . [3] (b) Consider the two models below: Model A: h c d t=+ , Model B: lnh c d t=+ , where c and d are constants. Explain which of Model A or Model B is a better fit to the data. State the equation of the least squares regression line in this case, giving your answer to 3 decimal places. [3] (c) Use the equation of your regression line in (b) to estimate the length of a 6-week-old bamboo stalk. Comment on the reliability of this estimate. [2] (d) Comment on the suitability of using this model in the long run. [1] (e) Let H denote the height of the bamboo stalk calculated for each value of t in the table using the least squares regression line obtained in (b). Find the value of S, where ( ) 2 S h H=− , giving your answer to 2 decimal places. [1] (f) For each of the eight sample values of t, 'H is given by ' lnH p q t=+ , where p and q are constants. Would ( ) 2 'hH− be greater or less than S and why? [1] t 1 2 3 4 5 7 8 9 h 3.6 7.4 10.2 11.7 12.6 14.1 14.5 15.5

7 ANGLO-CHINESE JUNIOR COLLEGE 2025 H2 MATHEMATICS 9758/02 [Turn over 11 In a laboratory, the time in minutes for an inspector to test an electrical circuit board is a continuous random variable X. The standard deviation of X is 0.63 and under ordinary conditions, the expected value of X is 5.82. As a result of the introdu